If $\vec{a} = 5\hat{i} - \hat{j} - 3\hat{k}$ and $\vec{b} = \hat{i} - 3\hat{j} - 5\hat{k}$ then $(\vec{a}+\vec{b}).(\vec{a}-\vec{b})$ is equal to:

If $\vec{a}$ is a non-zero vector, then always

Let $\vec{a},\vec{b},\vec{c}$ are three vectors such that $|\vec{a}|=3, |\vec{b}|=2, |\vec{c}|=1$. If the projection of $\vec{b}$ along $\vec{c}$ is equal to the projection of $\vec{a}$ along $\vec{c}$ and $\vec{b} \perp \vec{a}$ , then the value of $|2\vec{a}-2\vec{b}+3\vec{c}|$ is equal to

Which of the following statements are true?

(A) If $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$, then $x, y, z$ are called direction ratios of $\vec{r}$.

(B) For any two vectors $\vec{a}$ and $\vec{b}$, $\vec{a} + \vec{b} = \vec{b} + \vec{a}$

(C) $\vec{a} \perp \vec{b}$ if and only if $\vec{a} \times \vec{b} = \vec{0}$

(D) Projection of $\vec{b}$ on $\vec{a}$ is $\frac{\vec{a} \cdot \vec{b}}{|\vec{a}|^2}$

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Let $\vec{a} = 2\hat{i} - \hat{j}, \vec{b} =- 4\hat{j} + k\,\text{and}\,\vec{c} = \hat{i} + 2\hat{k}$. If $\vec{d}$ is a vector perpendicular to both $\vec{a}$ and $\vec{b}$ such that $\vec{c} \cdot \vec{d} = 34$, then $|\vec{d}|$ is equal to

If $\hat{i}$, $\hat{j}$ and $\hat{k}$ are unit vectors along the co-ordinate axes OX, OY and OZ respectively, then

(A) $\hat{i} \times \hat{j} = \hat{k}$

(B) $\hat{k} \times \hat{i} = -\hat{j}$

(C) $\hat{j} \cdot \hat{j} = 1$

(D) $\hat{j} \cdot \hat{k} = 0$

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Let $\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}$ and $\vec{b} = -2\hat{i} + 3\hat{j} - 4\hat{k}$, then which of the following statements are correct?

(A) $|\vec{a}| = \sqrt{14}$

(B) $|\vec{b}| = 29$

(C) $\vec{a} \cdot \vec{b} = 8$

(D) Angle between $\vec{a}$ and $\vec{b}$ is $\cos^{-1}\left(\frac{-8}{\sqrt{406}}\right)$

Choose the correct answer from the options given below: